Greek warriors did not have tattoos. In fact, Greek warriors would rather die than be tattooed. Literally: we have a grave epitaph from about 480 BC in which a man named Pollis, from Megara, declares that "not being a coward, I met my end at the hands of the tattooers." Pollis kept his pride by not letting anyone put ink on his skin.

Why were Greek fighters so fiercely opposed to tattoos? Because tattooing was something they did to enslaved people. Greeks (and Persians) put tattoos on slaves to punish them, especially if they had tried to escape. A few famous cases of enslaved war captives being tattooed drive home the point that free Greeks saw this as a horrific indignity. Herodotos tells us of the fate of the Thebans who stood with Leonidas at Thermopylai, but surrendered instead of fighting to the death:

When the barbarians took hold of them as they approached, they killed some of them as they came close. Most of them were tattooed by Xerxes' command with the king's marks, starting with the general Leontiades.

-- Hdt. 7.233.2

Plutarch claims that the same fate befell some of the hapless Athenians of the expeditionary force in Sicily when they were captured by the Syracusans in 413 BC - after the Athenians had themselves tattooed Syracusan captives:

Not a few were stolen away and sold into slavery, or succeeded in passing themselves off for serving men. These, when they were sold, were tattooed on the forehead with the mark of a horse—yes, there were some who actually had to suffer this in addition to being enslaved.

-- Plutarch, Life of Nikias 29.1

This was the fate that Pollis of Megara was trying to avoid. The tattoo would mark him forever as someone who had been captured in battle and sold into slavery. Repulsed by the thought, he chose to die fighting, and his relatives wanted him to be immortalised as a man who had done so.

Since tattoos were so strongly associated with slavery, tattooing was not developed as an art form among the Ancient Greeks, and all designs we know about from vases are very simple - a few lines or chevrons along the arm, or a rudimentary animal figure such as the Athenian owl or the Syracusan horse. In some cases we're not even sure if the scribbles on a vase painting of a person are meant to show tattoos, scars, or other marks. When we do see more elaborate motifs on vases, it is usually assumed that we are dealing with Thracian or Skythian enslaved people (typically women) who came from cultures where tattooing was more common and respected. They would have had their tattoos done before they were enslaved.

But this ethnic marker, in turn, makes it more likely that musicians would have tattoos. Perhaps the most common type of musician in the Greek world was the flute girl, a staple of elite drinking parties and a stock character in comedies. These were enslaved women trained to dance and play instruments and hired out as entertainers. Their fate is a grim one; elite symposia were raucous affairs involving much alcohol and few inhibitions, and the flute girls - who were forced to perform naked - were usually the only women present. At the beginning of the evening they may have been chiefly appreciated for their skills and performance, but it would not have been rare for these parties to end in sex between male guests and sexual violence targeted at the entertainers.

Because the flute girls were slaves, they may have been tattooed, either before or after their enslavement. If they did have tattoos, their nudity would have made it instantly clear that they were foreign, enslaved, and therefore socially dead and sexually available. It may have only increased their exotic appeal in the eyes of the elite men present, who were used to regarding female slaves as sexual objects as a matter of course.

In short, the only tattoo designs you would see on Greek warriors would be the simple marks of their enslavement. The designs seen on musicians would have been more diverse, especially if they were Thracian or Skythian slaves. But Greeks struggled to describe or depict these designs because they abhorred tattooing as an art form and associated it entirely with slavery. You're better off looking for the designs seen on Skythians, which are conformed archaeologically by the discovery of preserved skin in Siberian graves.